President Obama delivers a statement from the White House Rose Garden addressing the Affordable Care Act and the current government shutdown.

By Carrie Dann and Michael O'Brien, NBC News

As the federal government shutdown neared the end of its first business day, the House failed to pass a series of separate bills to restart funding for national parks, veterans' services and the city of Washington, D.C.

The measures, designed to eliminate some of the most unpopular consequences of the ongoing shutdown, required a two-thirds majority vote under the House's rules.

Democrats remained mostly united against the funding bills, which they argued amounted to the GOP's "cherry-picking" of politically palatable federal spending while ignoring the problems of the larger government funding lapse.

The vote on restoring funding for the Veterans Affairs Department was 264-164, with 33 Democrats voting in support. A measure to fund the city of D.C. failed by a similar margin. And a bill to keep national parks and museums open also failed to get the two-thirds majority needed.

Even if passed, the piecemeal continuing resolutions -- dubbed "mini-CRs"-- would have faced a quick demise.

Democratic leaders immediately rejected the plan earlier Tuesday, holding to their demand that Republicans relent and pass the six-week extension of government spending sent to them by the Senate.

"These piecemeal efforts are not serious, and they are no way to run a government," White House spokeswoman Amy Brundage said, adding, "The President and the Senate have been clear that they won't accept this kind of game-playing, and if these bills were to come to the President's desk he would veto them."

J. Scott Applewhite / AP

With the federal government out of money and out of time, House Majority Leader Eric Cantor of Va., third from left, meets with House Republican conferees as the Republican-controlled House and the Democrat-controlled Senate remain at an impasse, neither side backing down over Obamacare, Tuesday, Oct. 1, 2013 on Capitol Hill in Washington.

Senate Majority Leader Harry Reid, D-Nev., called the Republican effort a "wacky idea" that would simply "cherry-pick some of the few parts of government that they like."

"First, we need to end the government shutdown, and then Democrats are happy to agree on funding specific items," Reid said.

House Minority Leader Nancy Pelosi, D-Calif., said the GOP plan was akin to "releasing one hostage at a time."

A spokesman for House Speaker John Boehner called the White House's veto threat "unsustainably hypocritical."

"How does the White House justify signing the troop funding bill, but vetoing similar measures for veterans, National Parks, and District of Columbia?" said spokesman Michael Steel. "The President can't continue to complain about the impact of the government shutdown on veterans, visitors at National Parks, and DC while vetoing bills to help them."

Earlier, President Barack Obama continued to place blame for the shutdown squarely on the GOP.

"This Republican shutdown did not have to happen. But I want every American to understand why it did happen," Obama said during remarks in the Rose Garden. "They've shut down the government over an ideological crusade to deny affordable health insurance to millions of Americans."

Obama scolded the GOP for its focus on his signature domestic achievement, urging Congress to take up a simple continuation of government spending or risk the economic fallout that would come as a result.

"This is only going to happen when Republicans realize they don't get to hold the economy hostage over ideological demands," Obama said. "It's all about rolling back the Affordable Care Act. This, more than anything else, seems to be what the Republican Party stands for these days."

Republicans are currently trying to reopen popular parts of the government by offering money to fund entities such as national parks, but the Democrats are comparing that to a "release-one-hostage-at-a-time strategy," insisting the full government be funded rather than doing it one piece at a time. NBC's Kelly O'Donnell reports

Complicating matters for Obama, his Democratic allies and Republican adversaries in Congress is that they must now venture onto uncertain political terrain, given that this shutdown is the first one in 17 years.

The shutdown itself came to pass following a flurry of activity late Monday, which ended after Democrats rejected an 11th-hour attempt by Republicans to convene a “conference” committee – the formal process of resolving differences between House and Senate legislation – after their repeated attempts to both fund government and undo either part or all of Obamacare were rejected by the Democratic Senate.

On Tuesday, Reid said that Democrats would be happy to engage in a conference committee – but only after Republicans relented, and passed the extension of government spending through mid-November favored by Democrats.

“We will not go to conference with a gun to our head,” Reid said late Monday on the Senate floor.

In the absence of a solution, leaders in both parties shifted to playing the blame game, as Republicans accused Democrats of having forced the shutdown, and vice versa.

And new polling released Tuesday morning suggested that blame was likely to fall unevenly upon both political parties, with Americans directing their ire more toward Republicans in Congress than Obama or Democratic lawmakers.

A Quinnipiac University poll found that Americans gave Republicans in Congress their lowest marks ever, with 74 percent disapproving of the way the GOP is handling its job. (Sixty percent of Americans said they disapproved of the way Democrats in Congress were handling their job, and 49 percent disapproved of the way Obama is handling his job.)

Moreover, the poll also found that voters broadly oppose – 72 to 22 percent – shutting down the federal government to block the implementation of Obamacare, the core element of Republicans’ strategy to date.